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If you know the structure, maybe you can calculate the density from the ionic radii.
Calculate also for another similar material with a known density to check the method.
See the table here; http://en.wikipedia.org/wiki/Ionic_radius

It's pretty easy if you know the structure. If you don't then some sort of mass spec is your only hope on a real sample.

You basically need to find a repeating unit of structure then calculate the weight / volume of that repeating unit. On planar surfaces sometimes planar density is calculated as weight / area instead like graphene as it's thickness is pretty much non-existent.

If you have the chemical formula (which may not be consistent for a thin film if it is doped) you assume you have a repeating unit of the chemical formula calculate the percent mass of Nitrogen and multiply that by total mass/total volume. Similar to above, but easier.

Otherwise you do a mass spec of the results figure % mass nitrogen and divide by volume.